Group #30772213 — subgroups

Subgroups — #30772213

Order 17152.

310 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#22560373	C₂	1	1	yes	maximal, characteristic
3	#22558300	C₂	1	4	yes	maximal
4	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14523977	C₂²	1	2	yes	—
7	C₂×#14523975	—	2	1	no	—
8	#16863262	—	2	1	no	—
9	C₄×C₆₇⋊C₁₆	C₄	1	4	yes	complemented
10	C₂×#107×C₆₇	C₂²	1	2	yes	—
11	C₂×#106×C₆₇	—	2	1	no	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#382×C₆₇	—	2	1	no	—
14	C₂×C₄⋊Dic₆₇	—	2	2	no	—
15	C₂×C₆₇⋊C₁₆	—	2	2	no	—
16	C₂×C₆₇⋊C₁₆	—	2	2	no	—
17	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
18	#14523977	—	2	4	no	—
19	#14523975	—	4	2	no	—
20	C₆₇⋊#95	—	4	2	no	—
21	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
24	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
25	#107×C₆₇	—	2	4	no	—
26	#106×C₆₇	—	4	2	no	—
27	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₆₇	Q₈	1	2	yes	—
29	#95×C₆₇	—	4	2	no	—
30	C₂²×Dic₆₇	—	4	2	no	—
31	C₄⋊Dic₆₇	—	4	2	no	—
32	C₄⋊Dic₆₇	—	4	2	no	—
33	C₆₇⋊C₁₆	—	2	4	no	—
34	C₂²×C₄×C₆₇	—	4	2	no	—
35	C₂²×C₄×C₆₇	D₈	1	2	yes	—
36	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
38	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
39	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
40	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₈×C₆₇	#41	1	2	yes	—
43	C₂×C₈×C₆₇	#41	1	2	yes	—
44	C₄⋊C₄×C₆₇	—	4	2	no	—
45	C₄⋊C₄×C₆₇	—	4	2	no	—
46	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
47	C₄²×C₆₇	Q₁₆	1	2	yes	—
48	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₆₇	—	2	4	no	—
50	C₂×C₄×C₆₇	—	8	2	no	—
51	C₂×C₄×C₆₇	—	8	4	no	—
52	C₂×C₄×C₆₇	#107	1	4	yes	—
53	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
55	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
56	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₆₇	—	8	2	no	—
60	C₂×Dic₆₇	—	8	4	no	—
61	C₈×C₆₇	#106	1	4	yes	—
62	C₈×C₆₇	#117	1	2	yes	—
63	C₈×C₆₇	#125	1	2	yes	—
64	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
65	C₂²×C₆₇	#486	1	1	yes	characteristic
66	C₂²×C₆₇	#486	1	1	yes	characteristic
67	C₂²×C₆₇	#508	1	4	yes	—
68	C₄×C₆₇	—	2	4	no	—
69	C₄×C₆₇	—	16	2	no	—
70	C₄×C₆₇	#382	1	1	yes	characteristic
71	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
72	C₄×C₆₇	#487	1	1	yes	characteristic
73	C₄×C₆₇	#487	1	1	yes	characteristic
74	Dic₆₇	—	16	2	no	—
75	#12889	—	67	1	no	maximal, Sylow
76	C₂×C₆₇	#1834	1	1	yes	characteristic
77	C₂×C₆₇	#2158	1	2	yes	—
78	C₂×C₆₇	#2242	1	2	yes	—
79	C₂×C₆₇	#2242	1	2	yes	—
80	C₂×C₄×C₁₆	—	67	1	no	—
81	#1846	—	67	1	no	—
82	#1846	D₆₇	1	1	yes	characteristic
83	#1380	—	67	4	no	—
84	C₆₇	#12889	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₁₆	—	67	2	no	—
86	C₂×#107	—	67	2	no	—
87	C₂×#107	C₂×D₆₇	1	2	yes	—
88	C₂×#106	—	2	1	no	—
89	C₂×#106	—	134	1	no	—
90	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
91	#382	—	2	1	no	—
92	#382	—	134	1	no	—
93	C₄×C₁₆	—	67	4	no	—
94	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	134	2	no	—
98	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
99	C₂×C₁₆	—	67	4	no	—
100	C₂×C₁₆	—	134	2	no	—
101	C₂×C₁₆	—	134	2	no	—
102	#107	—	2	4	no	—
103	#107	—	134	4	no	—
104	#106	—	4	2	no	—
105	#106	—	268	2	no	—
106	C₄×C₈	Dic₁₃₄	1	2	yes	—
107	C₄×C₈	C₄×D₆₇	1	2	yes	—
108	#95	—	4	2	no	—
109	#95	—	268	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	—	268	2	no	—
112	C₂²×C₄	#11761613	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
114	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
115	C₂×C₈	#11761615	1	2	yes	—
116	C₂×C₈	#11761616	1	2	yes	—
117	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
118	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
119	C₂×C₈	#11761627	1	2	yes	—
120	C₂×C₈	#11761629	1	2	yes	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	268	2	no	—
124	C₄⋊C₄	—	268	2	no	—
125	C₄²	Dic₆₇⋊C₄	1	2	yes	—
126	C₄²	#11761618	1	2	yes	—
127	C₁₆	—	134	4	no	—
128	C₂³	C₂×#11761613	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	536	2	no	—
133	C₂×C₄	—	536	4	no	—
134	C₂×C₄	#14523976	1	4	yes	—
135	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
136	C₂×C₄	#14524030	1	1	yes	characteristic
137	C₂×C₄	#14524032	1	1	yes	characteristic
138	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
139	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
140	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
141	C₈	#14523974	1	4	yes	—
142	C₈	#14524001	1	2	yes	—
143	C₈	#14524034	1	2	yes	—
144	C₂²	C₂×#14523976	1	1	yes	characteristic
145	C₂²	#16863742	1	1	yes	characteristic
146	C₂²	#16863744	1	1	yes	characteristic
147	C₂²	#16863826	1	4	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	16	2	no	—
150	C₄	—	1072	2	no	—
151	C₄	#16863261	1	1	yes	characteristic
152	C₄	C₂×#14523974	1	1	yes	characteristic
153	C₄	#16863746	1	1	yes	characteristic
154	C₄	#16863748	1	1	yes	characteristic
155	C₂	#22560336	1	1	yes	characteristic
156	C₂	#22561991	1	2	yes	—
157	C₂	#22562440	1	2	yes	—
158	C₂	#22562443	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₆₇⋊C₁₆

C₂

yes

maximal, characteristic

#22560373

C₂

yes

maximal, characteristic

#22558300

C₂

yes

maximal

C₆₇×#1846

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

—

C₂×#14523977

C₂²

yes

—

C₂×#14523975

—

#16863262

—

C₄×C₆₇⋊C₁₆

C₄

yes

complemented

C₂×#107×C₆₇

C₂²

yes

—

C₂×#106×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#382×C₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

#14523977

—

#14523975

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

characteristic

#107×C₆₇

—

#106×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

#95×C₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

D₈

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

Q₁₆

yes

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#107

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

#106

yes

—

C₈×C₆₇

#117

yes

—

C₈×C₆₇

#125

yes

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#2242

yes

—

C₂×C₆₇

#2242

yes

—

C₂×C₄×C₁₆

—

#1846

—

#1846

D₆₇

yes

characteristic

#1380

—

C₆₇

#12889

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

C₂×D₆₇

yes

—

C₂×#106

—

C₂×#106

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#382

—

#382

—

134

—

C₄×C₁₆

—

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂²⋊D₆₇

yes

characteristic

C₂×C₁₆

—

100

C₂×C₁₆

—

134

—

101

C₂×C₁₆

—

134

—

102

#107

—

103

#107

—

134

—

104

#106

—

105

#106

—

268

—

106

C₄×C₈

Dic₁₃₄

yes

—

107

C₄×C₈

C₄×D₆₇

yes

—

108

#95

—

109

#95

—

268

—

110

C₂²×C₄

—

111

C₂²×C₄

—

268

—

112

C₂²×C₄

#11761613

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

114

C₂×C₈

Dic₆₇⋊C₄

yes

—

115

C₂×C₈

#11761615

yes

—

116

C₂×C₈

#11761616

yes

—

117

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

118

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

119

C₂×C₈

#11761627

yes

—

120

C₂×C₈

#11761629

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

268

—

124

C₄⋊C₄

—

268

—

125

C₄²

Dic₆₇⋊C₄

yes

—

126

C₄²

#11761618

yes

—

127

C₁₆

—

134

—

128

C₂³

C₂×#11761613

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

536

—

133

C₂×C₄

—

536

—

134

C₂×C₄

#14523976

yes

—

135

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

136

C₂×C₄

#14524030

yes

characteristic

137

C₂×C₄

#14524032

yes

characteristic

138

C₂×C₄

C₂×#11761615

yes

characteristic

139

C₂×C₄

C₂×#11761616

yes

characteristic

140

C₂×C₄

C₂×#11761618

yes

characteristic

141

C₈

#14523974

yes

—

142

C₈

#14524001

yes

—

143

C₈

#14524034

yes

—

144

C₂²

C₂×#14523976

yes

characteristic

145

C₂²

#16863742

yes

characteristic

146

C₂²

#16863744

yes

characteristic

147

C₂²

#16863826

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

1072

—

151

C₄

#16863261

yes

characteristic

152

C₄

C₂×#14523974

yes

characteristic

153

C₄

#16863746

yes

characteristic

154

C₄

#16863748

yes

characteristic

155

C₂

#22560336

yes

characteristic

156

C₂

#22561991

yes

—

157

C₂

#22562440

yes

—

158

C₂

#22562443

yes

—

	subgroup	G/H
center	C₂³	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1846	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1846	D₆₇
hypocenter	C₆₇	#12889
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#11761613

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1846

C₂

socle, prime radical

C₂³×C₆₇

C₂×D₈

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
9. C₄×C₆₇⋊C₁₆	C₄	semidirect
84. C₆₇	#12889	semidirect

H ≅ G/N

type

9. C₄×C₆₇⋊C₁₆

C₄

semidirect

84. C₆₇

#12889

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄×C₆₇⋊C₁₆	2
#22560373	2
#22558300	2
C₆₇×#1846	2
#12889	67

subgroup

index

C₂×C₄×C₆₇⋊C₁₆

#22560373

#22558300

C₆₇×#1846

#12889

Subgroups — #30772213

Conjugacy classes of subgroups

Complements

Maximal subgroups